Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

We give a necessary and sufficient condition for that the support τ -tilting poset of a finite dimensional algebra Λ is isomorphic to the poset of symmetric group S n+1 with weak order. Moreover we show that there are infinitely many finite dimensional algebras whose support τ -tilting posets are isomorphic to S n+1 .

A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Brüstle, G. Dupont and M. Pérotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. In [BDP], they considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type A or of type Ãn,1 by using theory of tilting mutation.

Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support τ -tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support τ -tilting posets coincide with those of tree quiver algebras and give a full characterization of such algebras., we call it a subposet of P if a ≤ P ′ b implies a ≤ b for any elements a, b of P ′ . A subposet P ′ of P is said to be full provided the converse of the implication above holds: i.e. the partial orders of P and P ′ coincide. Moreover, we say that a subposet P ′ of P is induced if it is full and the partial order ≤ induces that H(P ′ ) is a full subquiver of H(P).

Abstract. D.Happel and L.Unger defined a partial order on the set of basic tilting modules. We study the poset of basic pre-projective tilting modules over path algebra of infinite type. We give an equivalent condition for that this poset is a distributive lattice. We also give an equivalent condition for that a distributive lattice is isomorphic to the poset of basic pre-projective tilting modules over path algebra of infinite type.